We present the results from the timing and spectral study of Mrk 421 based mainly on the Swift data in the X-ray energy range obtained during the time interval 2015 December-2018 April. The most extreme X-ray flaring activity on the long-term, daily and intraday timescales was observed during the 2-month period which started in 2017 December when the 0.3-10 keV flux exceeded a level of 5×10 −9 erg cm −2 s −1 , recorded only twice previously. While the TeV-band and X-ray variabilities mostly were correlated, the source often varied in a complex manner in the MeV-GeV and radio-UV energy ranges, indicating that the multifrequency emission of Mrk 421 could not be always generated in a single zone. The longer-term flares at X-rays and γ-rays showed a lognormal character, possibly indicating a variability imprint of the accretion disk onto the jet. A vast majority of the 0.3-10 keV spectra were consistent with the log-parabolic model, showing relatively low spectral curvature and correlations between the different spectral parameters, predicted in the case of the first and secondorder Fermi processes. The position of the synchrotron spectral energy distribution (SED) peak showed an extreme variability on diverse timescales between the energies E p <0.1 keV and E p >15 keV, with 15% of the spectra peaking at hard X-rays and was related to the peak height as S p ∝E α p with α∼0.6, which is expected for the transition from Kraichnan-type turbulence into the "hard-sphere" one. The 0.3-300 GeV spectra showed the features of the hadronic contribution, jet-star interaction and upscatter in the Klein-Nishina regime in different time intervals.

Context. The multi-wavelength (MWL) properties of some TeV-detected blazars are still poorly understood. By studying the MWL properties of the BL Lacertae source 1ES 0033+595, we make further progress in our understanding of the instable physical processes responsible for particle acceleration to ultra-relativitic energies and the observed emission properties. Aims. This paper presents the results of a detailed X-ray timing and spectral analysis of the source 1ES 0033+595, allowing us to draw conclusions about the physical mechanisms responsible for particle acceleration and the generation of X-ray emission. We also examined the long-term MWL behaviour of the source and interband cross-correlations. Methods. Our study focused on the observations performed with the X-Ray Telescope on board the Neil Gehrels Swift Observatory (Swift-XRT) in the period 2005-2022. The MWL data sets were obtained with several space and ground-based instruments. We analysed the MWL flaring behaviour during the entire period, as well as in particular subperiods selected on the basis of the X-ray flaring activity of the target; constructed histograms characterising the distributions of the different spectral parameters and MWL fluxes; and checked cross-correlations between these parameters and/or MWL fluxes. The obtained results are compared with the predictions of the various theoretical studies and simulations presented by the different authors to date. Results. The source showed extreme spectral properties with dominance of high spectral curvature, frequent occurrence of very and extremely hard photon indices, and the presence of a synchrotron spectral energy distribution (SED) peak in the hard X-ray range (sometimes at energies beyond 10 keV). These properties reveal the importance of first-order Fermi acceleration with very low initial particle energy distribution, along with the co-existence of stochastic acceleration and hadronic processes. The source was characterised by very uneven and erratic flaring activity in diverse epochs: the period of strong flares (2013)(2014)(2015)(2016) was preceded by a moderate variability and followed by a gradual long-term decrease in MWL flaring activity. We detected a number of instances of intraday 0.3-10 keV variability, which were sometimes observed within a few hundred seconds and explained by the interaction between the relativistic shock front and jet inhomogeneities with strong magnetic fields. The X-ray and γ-ray fluxes showed a lognormal distribution, which hints at the imprint of accretion disc instabilities on the blazar jet.

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